Importance sampling method for multiple failure regions

ABSTRACT

A method of circuit yield analysis for evaluating rare failure events existing in multiple disjoint failure regions defined by a multi-dimensional parametric space, the method including performing initial sampling to detect failed samples respectively located at multiple failure regions in the multi-dimensional parametric space, performing clustering to identify the failure regions, performing feature filtering to determine which parameter component is a non-principal component in affecting circuit yield, applying a dimensional reduction method on a dimension corresponding to the parameter component, optimizing an importance sampling (IS) distribution function corresponding to each of the failure regions, and constructing a final importance sampling (IS) distribution function using a mixed Gaussian (mGaussian) function corresponding to all of the failure regions.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application claims priority to, and the benefit of, U.S.Provisional Application 62/395,334, filed on Sep. 15, 2016 in the U.S.Patent and Trademark Office, the entire content of which is incorporatedherein by reference.

FIELD

One or more aspects of embodiments according to the present inventiongenerally relate to a method for improving efficiency and accuracy ofimportance sampling Monte Carlo (ISMC) simulations by reducing a numberof simulations needed to identify and analyze multiple rare failureevents affecting product yield, and a system for performing the same.

BACKGROUND

In the field of integrated circuit (IC) yield analysis, it has becomeincreasingly challenging to evaluate very rare failure events (i.e.,rarely occurring failure events) when a more of process variabilitysources exist. Failure rates at “high-sigma” tails of a distribution(e.g., 6σ or higher) are important, as an array demands billions of lifecycles, and because failure of even only a few cells could becatastrophic. For example, for a 1 Mb memory block, to achieve yield of90%, each individual memory cell of the memory block may require afailure rate that is less than 1e−7. To ensure capture of an incrediblyrare failure event at a simulation-based evaluation/validation stage,more than 1e9 standard Monte Carlo (MC) simulations may be required.

SUMMARY

Aspects of embodiments of the present disclosure are directed toward amethod for improving importance sampling Monte Carlo simulation (ISMC)efficiency and accuracy, and toward a system for performing the same.

According to an embodiment of the present invention, there is provided amethod of circuit yield analysis for evaluating rare failure eventsexisting in multiple disjoint failure regions defined by amulti-dimensional parametric space, the method including performinginitial sampling to detect failed samples respectively located atmultiple failure regions in the multi-dimensional parametric space,performing clustering to identify the failure regions, performingfeature filtering to determine which parameter component is anon-principal component in affecting circuit yield, applying adimensional reduction method on a dimension corresponding to theparameter component, optimizing an importance sampling (IS) distributionfunction corresponding to each of the failure regions, and constructinga final importance sampling (IS) distribution function using a mixedGaussian (mGaussian) function corresponding to all of the failureregions.

The method may further include collecting the failed samples at eachfailure region, and calculating a final failure probability.

Performing initial sampling may include performing at least one ofuniform sampling, orthogonal sampling, Latin-Hypercube sampling, orSobol sampling.

Performing clustering may include performing at least one ofhierarchical clustering, K-means clustering, Expectation-Maximizationclustering, or density-based clustering.

Performing clustering may include using evaluation algorithms to scoreand determine a clustering strategy.

Using evaluation algorithms may include using at least one of aDavis-Bouldin index, a Dunn index, a Jaccard index, an F-measure, or aSilhouette coefficient.

Optimizing the importance sampling (IS) distribution functioncorresponding to each of the failure regions may include performing atleast one of sequential importance sampling, probability collectives, orregression-based sampling.

The method may further include parameterizing the mGaussian function.

Parameterizing the mGaussian function may include performing at leastone of closed-form calculations or convolutional calculations.

Applying the dimensional reduction method may include at least one ofprincipal-component-analysis or filtering.

According to another embodiment of the present invention, there isprovided a system for circuit yield analysis for evaluating rare failureevents existing in multiple disjoint failure regions defined by amulti-dimensional parametric space, the system including a processor,and a memory having instructions stored thereon that, when executed bythe processor, cause the processor to perform initial sampling to detectfailed samples respectively located at multiple failure regions in themulti-dimensional parametric space, perform clustering to identify thefailure regions, perform feature filtering to determine which parametercomponent is a non-principal component in affecting circuit yield, applya dimensional reduction method on a dimension corresponding to theparameter component, optimize an importance sampling (IS) distributionfunction corresponding to each of the failure regions, and construct afinal importance sampling (IS) distribution function using a mixedGaussian (mGaussian) function corresponding to all of the failureregions.

The instructions, when executed by the processor, may further cause theprocessor to collect the failed samples at each failure region, andcalculate a final failure probability.

The instructions, when executed by the processor, may cause theprocessor to perform the initial sampling by performing at least one ofuniform sampling, orthogonal sampling, Latin-Hypercube sampling, orSobol sampling.

The instructions, when executed by the processor, may cause theprocessor to perform the clustering by performing at least one ofhierarchical clustering, K-means clustering, Expectation-Maximizationclustering, or density-based clustering.

The instructions, when executed by the processor, may cause theprocessor to perform the clustering by using evaluation algorithms toscore and determine a clustering strategy.

The instructions, when executed by the processor, may cause theprocessor to optimize the importance sampling (IS) distribution functioncorresponding to each of the failure regions by performing at least oneof sequential importance sampling, probability collectives, orregression-based sampling.

The instructions, when executed by the processor, may further cause theprocessor to parameterize the mGaussian function.

The instructions, when executed by the processor, may cause theprocessor to parameterize the mGaussian function by performing at leastone of closed-form calculations or convolutional calculations.

The instructions, when executed by the processor, may cause theprocessor to apply the dimensional reduction method by at least one ofprincipal-component-analysis or filtering.

According to yet another embodiment of the present invention, there isprovided a method of circuit yield analysis for evaluating multipledisjoint failure regions defined by a multi-dimensional parametricspace, the method including identifying the multiple failure regions,and constructing a final importance sampling (IS) distribution functionusing a mixed Gaussian (mGaussian) function corresponding to all of thefailure regions.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other aspects of embodiments of the present invention will beappreciated and understood with reference to the specification, claims,and appended drawings wherein:

FIG. 1A depicts a uniform sampling of uniformly distributed samplesaccording to an importance sampling algorithm for detecting a failureregion corresponding to failed samples, according to an embodiment ofthe present invention;

FIG. 1B depicts a targeted sampling of non-uniformly distributed samplesfocused on the failure region detected by the uniform sampling of FIG.1A;

FIG. 2 depicts a flowchart of a method for constructing a mixed Gaussianfunction for focusing on multiple failure regions to improve importancesampling Monte Carlo simulations, according to an embodiment of thepresent invention;

FIG. 3A depicts a circuit diagram of a 6T-SRAM circuit;

FIG. 3B depicts a uniform sampling for detecting a failure region,according to an embodiment of the present invention;

FIG. 3C depicts a targeted sampling focused on the failure regiondetected by the uniform sampling of FIG. 3B;

FIG. 3D depicts a comparison of convergence resulting from standardMonte Carlo simulations, and convergence resulting from importancesampling Monte Carlo simulations, according to an embodiment of thepresent invention;

FIG. 4 depicts identification of possible failure regions for differentvariables corresponding to the 6T-SRAM circuit of FIG. 3 as identifiedusing the method of FIG. 2, according to an embodiment of the presentinvention;

FIG. 5 depicts benefits achieved by clustering using the method of FIG.2, according to an embodiment of the present invention; and

FIG. 6 depicts benefits achieved by feature filtering using the methodof FIG. 2.

DETAILED DESCRIPTION

Aspects of embodiments of the present disclosure are directed toward amethod for improving importance sampling Monte Carlo simulation (ISMC)efficiency and accuracy, and toward a system for performing the same.

Features of the inventive concept and methods of accomplishing the samemay be understood more readily by reference to the following detaileddescription of embodiments and the accompanying drawings. Hereinafter,example embodiments will be described in more detail with reference tothe accompanying drawings, in which like reference numbers refer to likeelements throughout. The present invention, however, may be embodied invarious different forms, and should not be construed as being limited toonly the illustrated embodiments herein. Rather, these embodiments areprovided as examples so that this disclosure will be thorough andcomplete, and will fully convey the aspects and features of the presentinvention to those skilled in the art. Accordingly, processes, elements,and techniques that are not necessary to those having ordinary skill inthe art for a complete understanding of the aspects and features of thepresent invention may not be described. Unless otherwise noted, likereference numerals denote like elements throughout the attached drawingsand the written description, and thus, descriptions thereof will not berepeated. In the drawings, the relative sizes of elements, layers, andregions may be exaggerated for clarity.

It will be understood that, although the terms “first,” “second,”“third,” etc., may be used herein to describe various elements,components, regions, layers and/or sections, these elements, components,regions, layers and/or sections should not be limited by these terms.These terms are used to distinguish one element, component, region,layer or section from another element, component, region, layer orsection. Thus, a first element, component, region, layer or sectiondescribed below could be termed a second element, component, region,layer or section, without departing from the spirit and scope of thepresent invention.

Spatially relative terms, such as “beneath,” “below,” “lower,” “under,”“above,” “upper,” and the like, may be used herein for ease ofexplanation to describe one element or feature's relationship to anotherelement(s) or feature(s) as illustrated in the figures. It will beunderstood that the spatially relative terms are intended to encompassdifferent orientations of the device in use or in operation, in additionto the orientation depicted in the figures. For example, if the devicein the figures is turned over, elements described as “below” or“beneath” or “under” other elements or features would then be oriented“above” the other elements or features. Thus, the example terms “below”and “under” can encompass both an orientation of above and below. Thedevice may be otherwise oriented (e.g., rotated 90 degrees or at otherorientations) and the spatially relative descriptors used herein shouldbe interpreted accordingly.

It will be understood that when an element, layer, region, or componentis referred to as being “on,” “connected to,” or “coupled to” anotherelement, layer, region, or component, it can be directly on, connectedto, or coupled to the other element, layer, region, or component, or oneor more intervening elements, layers, regions, or components may bepresent. In addition, it will also be understood that when an element orlayer is referred to as being “between” two elements or layers, it canbe the only element or layer between the two elements or layers, or oneor more intervening elements or layers may also be present.

In the following examples, the x-axis, the y-axis and the z-axis are notlimited to three axes of a rectangular coordinate system, and may beinterpreted in a broader sense. For example, the x-axis, the y-axis, andthe z-axis may be perpendicular to one another, or may representdifferent directions that are not perpendicular to one another.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the presentinvention. As used herein, the singular forms “a” and “an” are intendedto include the plural forms as well, unless the context clearlyindicates otherwise. It will be further understood that the terms“comprises,” “comprising,” “includes,” and “including,” when used inthis specification, specify the presence of the stated features,integers, steps, operations, elements, and/or components, but do notpreclude the presence or addition of one or more other features,integers, steps, operations, elements, components, and/or groupsthereof. As used herein, the term “and/or” includes any and allcombinations of one or more of the associated listed items. Expressionssuch as “at least one of,” when preceding a list of elements, modify theentire list of elements and do not modify the individual elements of thelist.

As used herein, the term “substantially,” “about,” and similar terms areused as terms of approximation and not as terms of degree, and areintended to account for the inherent deviations in measured orcalculated values that would be recognized by those of ordinary skill inthe art. Further, the use of “may” when describing embodiments of thepresent invention refers to “one or more embodiments of the presentinvention.” As used herein, the terms “use,” “using,” and “used” may beconsidered synonymous with the terms “utilize,” “utilizing,” and“utilized,” respectively. Also, the term “exemplary” is intended torefer to an example or illustration.

When a certain embodiment may be implemented differently, a specificprocess order may be performed differently from the described order. Forexample, two consecutively described processes may be performedsubstantially at the same time or performed in an order opposite to thedescribed order.

The electronic or electric devices and/or any other relevant devices orcomponents according to embodiments described herein may be implementedutilizing any suitable hardware, firmware (e.g. an application-specificintegrated circuit), software, or a combination of software, firmware,and hardware. For example, the various components of these devices maybe formed on one integrated circuit (IC) chip or on separate IC chips.Further, the various components of these devices may be implemented on aflexible printed circuit film, a tape carrier package (TCP), a printedcircuit board (PCB), or formed on one substrate. Further, the variouscomponents of these devices may be a process or thread, running on oneor more processors, in one or more computing devices, executing computerprogram instructions and interacting with other system components forperforming the various functionalities described herein. The computerprogram instructions are stored in a memory which may be implemented ina computing device using a standard memory device, such as, for example,a random access memory (RAM). The computer program instructions may alsobe stored in other non-transitory computer readable media such as, forexample, a CD-ROM, flash drive, or the like. Also, a person of skill inthe art should recognize that the functionality of various computingdevices may be combined or integrated into a single computing device, orthe functionality of a particular computing device may be distributedacross one or more other computing devices without departing from thespirit and scope of the described embodiments.

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which the present invention belongs. Itwill be further understood that terms, such as those defined in commonlyused dictionaries, should be interpreted as having a meaning that isconsistent with their meaning in the context of the relevant art and/orthe present specification, and should not be interpreted in an idealizedor overly formal sense, unless expressly so defined herein.

Due to the high number of MC simulations that may be required fordetection of incredibly rare failure events, importance sampling (IS)algorithms have been proposed and extensively studied for the purpose ofreducing a total number of MC simulations needed to detect the rarefailure events. The intent behind such studies is to shift and reshape,or modify, a distribution function (e.g., a Gaussian function) to movedeeply into a failure region corresponding to, or encompassing, theserarely occurring failure events, thereby largely reducing varianceoccurring within an MC simulation for a given number of random draws.Because an IS distribution function includes information related to thecause of deviation from the original distribution function (e.g., bycalculating a corresponding weight), the rare failure events occurringat a low rate, thus, can be evaluated within standard MC simulationtimes in accordance with a standard MC simulation budget (e.g., within astandard number of MC simulations).

Accordingly, a sampling distribution function may be modified from anoriginal distribution function to shift the mean value, and to reshapethe original distribution function to thereby change the sigma (□)(i.e., to reshape the original distribution function toward/into thefailure region), thereby placing a higher percentage of the samples atone or more failure corners/failure regions.

For example, one or more failure regions may be initially detected, andthen the original distribution function may be shifted and reshapedtoward the failure region(s) such that a larger number of samples at thefailure region(s) can be detected in a same number of simulations.Hence, more accuracy may be achieved despite using the same simulationbudget (i.e., despite running the same number of MC simulations), due tomore samplings being focused in the failure region(s), thereby reducingsimulation variance from MC simulations.

When MC simulation is used as a method of sampling, as a method ofaveraging, or as an integration method, some variance results due to therandom nature of MC simulations. According to example embodiments of thepresent invention, variance may be reduced and accuracy may be improvedat very low failure rates with a comparable, or even reduced, simulationbudget.

According to example embodiments of the present invention, a “relatedart” probability-collective based IS algorithm is used as shown below.The IS algorithm and the application thereof is known to those ofordinary skill in the art.

First, N₁ uniformly distributed samples ξ^(j)=(ξ₁ ^(j), . . . , ξ_(M)^(j)) are drawn from a given distribution function N(μ, σ), and MCsimulations are run to identify samples occurring in failure regions(e.g., to identify failure samples) and to calculate the L2-norm valuesof the failure samples with reference to the mean. Then, one of thefailed samples that has a minimum L2-norm is chosen as used as aninitial shift vector of μ¹*σ¹.

Thereafter, N₂ samples are drawn from an initial parameterizeddistribution h(ξ, μ¹, σ¹), and the iteration index may be set at 2(e.g., t=2).

Then, an indicator function I(ξ^(j)) may be evaluated with the N₂samples. Thereafter, the mean(s) and the sigma(s) may be calculatedthroughout the multiple dimensions of a parameterized spacecorresponding to the samples. The mean(s) and the sigma(s) mayrespectively be calculated by the following equations:

${\mu_{i}^{t} = \frac{\sum\limits_{j = 1}^{N}\;{{I\left( \xi^{j} \right)} \cdot {\omega\left( {\xi^{j},\mu_{i}^{t - 1},\sigma_{i}^{t - 1}} \right)} \cdot \xi^{j}}}{\sum\limits_{j = 1}^{N}\;{{I\left( \xi^{j} \right)} \cdot {\omega\left( {\xi^{j},\mu_{i}^{t - 1},\sigma_{i}^{t - 1}} \right)}}}};{and}$$\sigma_{i}^{t} = \sqrt{\frac{\sum\limits_{j = 1}^{N}\;{{I\left( \xi^{j} \right)} \cdot {\omega\left( {\xi^{j},\mu_{i}^{t - 1},\sigma_{i}^{t - 1}} \right)} \cdot \left( {\xi^{j} - \mu_{i}^{t}} \right)^{2}}}{\sum\limits_{j = 1}^{N}\;{{I\left( \xi^{j} \right)} \cdot {\omega\left( {\xi^{j},\mu_{i}^{t - 1},\sigma_{i}^{t - 1}} \right)}}}}$

Then, another batch of N2 samples may be drawn from the updatedparameterized distribution, and the iteration index, t, may be set ast+1 until convergence occurs (e.g., when the means and the sigmas arewithin an error tolerance).

Finally, N₃ samples may be drawn from the obtained optimal, or improved,sampling distribution h(ξ, {circumflex over (μ)}, {circumflex over(σ)}), and MC simulations may be run to identify failure samples. Thefailure probability may be expressed by the following equation:

$P_{r} = {\frac{1}{N_{3}}{\sum\limits_{j = 1}^{N_{3}}\;{{I\left( \xi^{j} \right)} \cdot \frac{\prod\limits_{i = 1}^{M}\;{h\left( \xi_{i}^{j} \right)}}{\prod\limits_{i = 1}^{M}\;{h\left( {\xi_{i}^{j},\hat{\mu},\hat{\sigma}} \right)}}}}}$

However, although embodiments of the present invention utilize ISalgorithms for circuit yield analysis, it should be noted that thepresent invention is not limited to the IS algorithm shown above. Inother embodiments, improvement of the IS algorithms may be achieved byother existing methods known to those skilled in the art, such asregression-based sampling and/or sequential importance sampling, etc.

FIG. 1A depicts a uniform sampling of uniformly distributed samplesaccording to an importance sampling algorithm for detecting a failureregion corresponding to failed samples, according to an embodiment ofthe present invention, and FIG. 1B depicts a targeted sampling ofnon-uniformly distributed samples focused on the failure region detectedby the uniform sampling of FIG. 1A.

Referring to FIGS. 1A and 1B, for example, by comparing the sampledistributions of FIG. 1A and FIG. 1B, it can be shown that by focusingthe simulations at a failure region 110 at the upper left quadrant(i.e., second quadrant), as shown in FIG. 1B, a larger number of failuresamples (e.g., higher percentage of samples) are detected at the failureregion in the graph of FIG. 1B than in the graph of FIG. 1A. This may beachieved by shifting the mean 120 from the center 130, as shown in FIG.1A, toward the failure region 110.

FIG. 2 depicts a flowchart of a method for constructing a mixed Gaussianfunction for focusing on multiple failure regions to improve importancesampling Monte Carlo simulations, according to an embodiment of thepresent invention.

Referring to FIG. 2, and also in reference to the importance sampling(IS) algorithms discussed above, although there are multiple differentIS methods, embodiments of the present invention introduce improved ISmethods, and also introduce methods for improving redistribution orresampling functions (e.g., Gaussian redistribution/resamplingfunctions) to generate a modified redistribution/resampling function.Although uniform sampling is described in the present embodiment, otherforms of sampling, such as orthogonal sampling, Latin-Hypercubesampling, or Sobol sampling, may instead be used to detect one or morefailure regions. According to the IS algorithm discussed above, auniform sampling is performed to draw, or sample, N1 uniformlydistributed samples (e.g., see FIG. 1A).

At S201, a uniform sampling of Monte Carlo simulations (i.e., a randomlysampling not limited to any area in the parametric space defined) may beperformed to detect failed samples. However, uniform sampling may not bepractical (e.g., because of unacceptably high variance or anunacceptable large number of variations) for an IC yield analysisinvolving multiple failure regions. However, at S201, because uniformlydistributed samples cover an entirety of a corresponding space (orregions), failure regions or corners (i.e., regions or corners at whichfailed samples are mainly located) may be detected.

Optionally, at S201.1, the uniform sampling may be combined withadvanced MC techniques, such as classifiers, statistical blockade orMarkov Chain MC (MCMC) methods, etc.

At S202, feature filtering may be performed to handle high-dimensionalproblems by determining which variables/parameters are non-principalcomponents (NPCs) in affecting the circuit yield, thereby enabling thesuppression of the impact of such variables by using dimensionalreduction techniques to reduce the MC variance at high dimensions withlimited sampling events during optimization/improvement of acorresponding IS distribution function. That is, suppression of theimpact of the non-principal components may be thought of as a “dimensionreduction” technique to enable determination of which variables arenon-principal components in affecting the circuit yield, so that theimpacts of these variables on the analysis may be suppressed during theimprovement of the IS distribution function. Accordingly, the featurefiltering allows for efficient analysis of a high-dimension problemhaving very low variance.

At S203, clustering algorithms may be used to group different, separatefailed samples into respective disjoint failure regions. That is, atS201, sometimes it may be apparent after drawing N1 uniformlydistributed samples that, instead of a single failure region, multiple,distinct failure regions exist (e.g., see the multiple failure regionsshown in FIG. 5, discussed further below). However, the detection ofmultiple failure regions indicates that a corresponding number ofGaussian distributions may be needed to cover all of the detectedfailure regions (e.g., one Gaussian distribution function per failureregion). To “contourize” the Gaussian function, the present embodimentprovides techniques for automatically determining methods for adjustingthe Gaussian function, and methods for optimizing/improving the Gaussiandistribution for each failure region, even for a sample having a largenumber of variables (i.e., very high dimensions in the parametric space)to allow for more rapid convergence of mean and sigma values despitegrowing variance.

In the present embodiment, a clustering method may be used, as opposedto a classification method, as resources need not be spent to generateboundaries between the successful samples and the failed samples (e.g.,to isolate the failure regions) in a clustering method. For example, inmachine learning, or statistical learning, upon detection of the failureregions, a classification approach requires generation of a function forclassifying, or distinguishing, failed samples and successful samples.Such an approach may require a relatively high level of effort togenerate an accurate boundary to identify failure regions. Additionally,a scoring strategy may be used to evaluate which clustering algorithm issuitable, and what number of failure regions should be individuallyidentified.

In IC yield analysis, due to the existence of multiple devices (e.g. theexistence of 6 transistors in an SRAM cell, as depicted in FIG. 3A), anddue to IC topology (e.g. pull-down (PD), pull-up (PU), and pass-gate(PG) pairs, some of which being depicted in FIG. 4), the multiplefailure regions may often be disconnected from each other in the highvariable/multi-dimensional parametric space. The embodiments describedherein overcome limitations of existing ISMC algorithms, which lack theefficiency and accuracy achieved herein. These limitations may beovercome using unsupervised learning/clustering algorithms at S203.Self-learning algorithms that have been adopted in the machine-learningcommunity may also be applied.

For example, a K-Means clustering algorithm may partition N samples X∈

^(N×M) into K disjoint clusters in which each sample belongs to thecluster with the nearest mean. K-Means clustering may be achieved byfirst determining the value K, and by randomly initializing clustercentroids μ₁ . . . μ_(k)∈

^(M). This process may be repeated at a t^(th) iteration, and for eachi, the equation may be set as c^(i)=arg min_(j)∥x^(i)−μ_(j)∥², while foreach j, the equation may be set as

$\mu_{i}^{t + 1} = \frac{\sum\limits_{i = 1}^{N}\;{{I\left( {c^{i} = j} \right)} \cdot x^{i}}}{\sum\limits_{i = 1}^{N}\;{I\left( {c^{i} = j} \right)}}$until convergence occurs (i.e. μ_(i) ^(t) when are within the errortolerance).

As part of the K-Means clustering algorithms used in this invention, theoptimal value of K may be determined by some scoring strategy, and the“Silhouette” score is utilized herein to measure how similar each sampleis to its own cluster when compared to the degree of similarity of othersamples to their respective cluster, thereby allowing a determination ofwhether the current K-means clustering strategy is optimal. In thepresent example, the equation

${S(i)} = \frac{{b(i)} - {a(i)}}{\max\left\{ {{a(i)} - {b(i)}} \right\}}$may be used, where b(i) is a lowest average dissimilarity of sample i toany other cluster of which i is not a member, and where a(i) is anaverage dissimilarity of sample i to its assigned cluster. According tothe above definition, S(i)∈[0,1].

Furthermore, it should be noted that other existing clusteringalgorithms may be used in embodiments of the present invention, such asExpectation Maximization (EM), hierarchical clustering, density-basedspatial clustering of applications with noise (DBSCAN), one-classSupport Vector Machine (SVM), and Artificial Neural Network (ANN), etc.Additionally, evaluation algorithms, such as a Davis-Bouldin index, aDunn index, a Jaccard index, an F-measure, or a Silhouette coefficient,may be used to score the clustering algorithms.

At S204, upon detection of the failure regions, the IS distributionfunction for each individual failure cluster may be improved (e.g.,individually). For example, mathematical algorithms may be used toimprove/optimize each IS function distribution function based oninformation corresponding to a respective failure region. For example, amean value of the sample distribution algorithm, as well as a sigmavalue of the sample distribution algorithm, may be updated based on thedetected failure regions. Accordingly, through an iterative process, animproved/optimal distribution function/sampling function may begenerated. During the iterative process, N2 samples are drawn or sampleduntil both mean and sigma values converge to be within an errortolerance. Furthermore, probability collectives (PC) and/or minimum L1/2norms may be used to improve or optimize the IS distribution functionfor each disjoint failure region.

Until convergence occurs, the method of the present embodiment mayrepeat S204 to further improve the IS function for each of the clustersof failed samples (e.g., each of the failure regions) determined atS203. Accordingly, S202, S203, and S204 provide a novel methodology tohandle the “multiple-failure-region” problems existing in IC yieldanalysis.

After applying the clustering algorithms (e.g. K-means, as mentionedlater) to optimize the mean and sigma values for each failure cluster,the final resampling function can be expressed as: g({right arrow over(ξ)})=Σ_(i=1) ^(K) w_(i)N({right arrow over (μ_(i))},{right arrow over(σ_(i))}) where {circumflex over (μ)}, {circumflex over (σ)}∈

^(K×M) are the mean and sigma tensors for failed samples at each clusterand w_(i)s follow Σ_(i=1) ^(K) w_(i)=1 are the normalized clusterweights that can be calculated as

$w_{i} = {\frac{1}{Z}{\int_{\overset{\rightarrow}{\mu_{i}} - {\alpha\overset{\rightarrow}{\sigma_{i}}}}^{\overset{\rightarrow}{\mu_{i}} + {\alpha\overset{\rightarrow}{\sigma_{i}}}}{{N\left( {\overset{\rightarrow}{\mu_{0}},\overset{\rightarrow}{\sigma_{0}}} \right)}{N\left( \ {\overset{\rightarrow}{\mu_{i}},\overset{\rightarrow}{\sigma_{i}}} \right)}d\overset{\rightarrow}{\xi}}}}$

At S205, upon convergence of the mean and sigma values, a mixture ofGaussian distributions (mGaussian distributions) may be constructed as afinal IS sampling function, such that parameters appearing in the mixedGaussian (mGaussian) function can be well determined by convolutionalmethods using the novel methodology for handlingmultiple-failure-regions provided herein. That is, once each of thedifferent failure regions has been identified, or labeled, then insteadof a single Gaussian distribution function for each failure region, amixture of Gaussians (mGaussian) will be collectively used as a singleIS sampling distribution function, wherein each Gaussian function may beimproved using probability collectives or regression models, etc. Forexample, an automated method may be used to construct the mGaussianfunction to provide a closed-form loop encompassing all of theindividual failure regions (e.g., based on convolution calculationwithout user input).

At S206, which may be a final integration step, once a sufficient numberof iterations are performed such that sufficient information is obtainedwith respect to the one or more failure regions to enable constructionof a single suitable mGaussian function covering multiple failureregions, the most recently obtained mean and sigma values may be used toperform MC simulations using the improved sample distribution toidentify the failure samples. Accordingly, even with a fixed simulationbudget (e.g., a simulation budget using the original distribution orsampling function), a higher accuracy level is achieved. It shouldfurther be noted that the methods of the described embodiments arecompatible with existing ISMC and other “high-sigma” yield evaluationmethods.

Suitable sample sizes N1, N2, and N3 may be selected by user input toadjust the sample sizes to suit specifications of the simulation.Although the present invention is not limited thereto, the sample sizesN1 and N3 may be larger than the sample size N2 because, for example,convergence may require a different number of loops or iterations.Accordingly, a user may select some criterion or criteria for theconvergence of the mean and sigma values.

FIG. 3A depicts a circuit diagram of a 6T-SRAM circuit, FIG. 3B depictsa uniform sampling for detecting a failure region, according to anembodiment of the present invention, FIG. 3C depicts a targeted samplingfocused on the failure region detected by the uniform sampling of FIG.3B, and FIG. 3D depicts a comparison of convergence resulting fromstandard Monte Carlo simulations, and convergence resulting fromimportance sampling Monte Carlo simulations, according to an embodimentof the present invention.

Referring to FIG. 3A, an example of a circuit 300 having 6 transistors310 each corresponding to a variable analyzed for determining thefailure regions. For example, improved ISMC methods may be performedduring 6T-SRAM yield analysis on the 6T-SRAM circuit 300 shown in FIG.3A. It should be noted that two-dimensional analysis corresponding toonly two of the transistors 310 is shown in FIGS. 3B and 3C for thepurpose of illustration, although the 6T-SRAM yield analysis may beperformed in 6-dimensional variable space (i.e., a variable spacedimension of six).

Referring to FIG. 3B a single failure region 320 (e.g., a singlecluster) is detected by performing uniform sampling (e.g., in accordancewith S201 described with respect to FIG. 2 above). In the presentexample, the samples are not uniform due to the existence of the failureregion 3250 that is detected primarily in a first quadrant, or upperright corner of FIG. 3B.

Referring to FIG. 3C, a probability collective method may be performed(e.g., in accordance with S202, S203, and S204 described with respect toFIG. 2 above) to improve the IS distribution function such that a higherpercentage of the samples are failure samples. That is, by improving theredistribution function to be further in the failure region 320, adenser amount of samples are placed in the failure region 320. Thesimulations may include, for example, BSIM-4 models for transistors at20 LP, VDD=0.90V, and Bitline capacitance of 0.1 pF. Additionally, andfor example, the variability may assume Vth variability, such thatσVt,PD=25 mV, σVt,PU=30 mV, and σVt,PG=30 mV.

Referring to FIG. 3D, the improved IS function may be used to performMonte Carlo simulations (e.g., in accordance with S206 described withrespect to FIG. 2 above). It should be noted that FIG. 3D also depictsconvergence as a result of standard Monte Carlo simulations 330 forcomparison. As shown, the improved PCIS-MC simulations 340 use fewersimulations than the Standard Monte Carlo simulations 330 to converge tothe final failure mark indicated by the dashed line 350.

FIG. 4 depicts identification of possible failure regions for differentvariables corresponding to the 6T-SRAM circuit of FIG. 3 as identifiedusing the method of FIG. 2, according to an embodiment of the presentinvention.

Referring to FIG. 4, for some specification changes of the 6T-SRAMcircuit evaluation, a critical problem may be a result of mismatch dueto symmetry of SRAM circuit topology (e.g., two left inverters, and tworight inverters). If a mismatch between a pair of the inverters occurs,convergence of the IS function at only one of the failure regions mayignore a separate failure region, thereby resulting in erroneousanalysis. However, the clustering algorithms discussed above withrespect to S203 ensure that all failure regions are considered. Forexample, with reference to the STATIC HOLD and STATIC READ depicted inFIG. 4, the optimal number of clusters is identified as being equal to2, in accordance with embodiments of the present invention.

FIG. 5 depicts benefits achieved by clustering using the method of FIG.2, according to an embodiment of the present invention.

Referring to FIG. 5, the clustering algorithms discussed with respect toS203 of FIG. 2 above may account for the SRAM read margin, for example,such that the read margin may be measured by minimum squares of an upperbranch 510 and a lower branch 520. By creating a mismatch such thateither one will fail to generate the shown failure pattern, byperforming the clustering algorithm to detect two clusters, and byscoring the clusters, two Gaussian functions may be used to respectivelysample the two clusters.

Accordingly, as discussed with respect to S205 of FIG. 2 above, the twoindividual Gaussian functions may be integrated to allow formultiple-failure-region importance sampling using a single mGaussianfunction. As can be seen in FIG. 5, the standard ISMC 530 (singlecluster) will converge at a rate that is different than the convergencerate of the multiple-failure-region ISMC 540, although both willconverge more quickly than the standard Monte Carlo simulation 550.

FIG. 6 depicts benefits achieved by feature filtering using the methodof FIG. 2.

Referring to FIG. 6, in the present example, two variables are projectedfor ease of description, although there is a total of six variablesrespectively associated with the transistors of the 6T-SRAM circuit.Further, in the present example, all of the samples result from the samefailure device with the different failure regions being projected intodifferent spaces and different dimensions.

Further, it can be seen in the failure regions 610 of FIG. 6 that thedistribution of failure samples 620 occurs in a large section of a rangeof the variables. Accordingly, the variables are normalized at zero suchthat zero is the mean. Additionally, because the failure samples may bedistributed almost uniformly over an entirety of the variable range, itmay be deduced that the corresponding variable (e.g., variable PD2 630)is not a limiting factor of the failure rate. However, by projecting thesame failure samples into PD1 and PU1, it appears that the failuresamples occupy a particular failure region, indicating a relativelysmall failure region to be detected for further analysis.

Upon determining which variable/dimension is critical, and by applying afeature filtering/dimension reduction technique for those unimportant,non-critical variables (e.g., in accordance with S202 described above),the reference IS may be improved enabling convergence with feweriterations.

The foregoing is illustrative of example embodiments, and is not to beconstrued as limiting thereof. Although a few example embodiments havebeen described, those skilled in the art will readily appreciate thatmany modifications are possible in the example embodiments withoutmaterially departing from the novel teachings and advantages of exampleembodiments. Accordingly, all such modifications are intended to beincluded within the scope of example embodiments as defined in theclaims. In the claims, means-plus-function clauses are intended to coverthe structures described herein as performing the recited function andnot only structural equivalents but also equivalent structures.Therefore, it is to be understood that the foregoing is illustrative ofexample embodiments and is not to be construed as limited to thespecific embodiments disclosed, and that modifications to the disclosedexample embodiments, as well as other example embodiments, are intendedto be included within the scope of the appended claims. The inventiveconcept is defined by the following claims, with equivalents of theclaims to be included therein.

What is claimed is:
 1. A method of circuit yield analysis for evaluatingrare failure events occurring in an integrated circuit design andexisting in multiple disjoint failure regions defined by amulti-dimensional parametric space, the method comprising: performinginitial sampling to detect failed samples respectively located atmultiple failure regions in the multi-dimensional parametric space;performing clustering to identify the failure regions; performingfeature filtering to determine which parameter component is anon-principal component in affecting circuit yield; applying adimensional reduction method on a dimension corresponding to theparameter component; optimizing an importance sampling (IS) distributionfunction corresponding to each of the failure regions; and constructinga final importance sampling (IS) distribution function using a mixedGaussian (mGaussian) function corresponding to all of the failureregions containing the rare failure events.
 2. The method of claim 1,further comprising: collecting the failed samples at each failureregion; and calculating a final failure probability.
 3. The method ofclaim 1, wherein performing initial sampling comprises performing atleast one of uniform sampling, orthogonal sampling, Latin-Hypercubesampling, or Sobol sampling.
 4. The method of claim 1, whereinperforming clustering comprises performing at least one of hierarchicalclustering, K-means clustering, Expectation-Maximization clustering, ordensity-based clustering.
 5. The method of claim 1, wherein performingclustering comprises using evaluation algorithms to score and determinea clustering strategy.
 6. The method of claim 5, wherein usingevaluation algorithms comprises using at least one of a Davis-Bouldinindex, a Dunn index, a Jaccard index, an F-measure, or a Silhouettecoefficient.
 7. The method of claim 1, wherein optimizing the importancesampling (IS) distribution function corresponding to each of the failureregions comprises performing at least one of sequential importancesampling, probability collectives, or regression-based sampling.
 8. Themethod of claim 1, further comprising parameterizing the mGaussianfunction.
 9. The method of claim 8, wherein parameterizing the mGaussianfunction comprises performing at least one of closed-form calculationsor convolutional calculations.
 10. The method of claim 1, whereinapplying the dimensional reduction method comprises at least one ofprincipal-component-analysis or filtering.
 11. A system for circuityield analysis for evaluating rare failure events occurring in anintegrated circuit design and existing in multiple disjoint failureregions defined by a multi-dimensional parametric space, the systemcomprising: a processor; and a memory having instructions stored thereonthat, when executed by the processor, cause the processor to: performinitial sampling to detect failed samples respectively located atmultiple failure regions in the multi-dimensional parametric space;perform clustering to identify the failure regions; perform featurefiltering to determine which parameter component is a non-principalcomponent in affecting circuit yield; apply a dimensional reductionmethod on a dimension corresponding to the parameter component; optimizean importance sampling (IS) distribution function corresponding to eachof the failure regions; and construct a final importance sampling (IS)distribution function using a mixed Gaussian (mGaussian) functioncorresponding to all of the failure regions containing the rare failureevents.
 12. The system of claim 11, wherein the instructions, whenexecuted by the processor, further cause the processor to: collect thefailed samples at each failure region; and calculate a final failureprobability.
 13. The system of claim 11, wherein the instructions, whenexecuted by the processor, cause the processor to perform the initialsampling by performing at least one of uniform sampling, orthogonalsampling, Latin-Hypercube sampling, or Sobol sampling.
 14. The system ofclaim 11, wherein the instructions, when executed by the processor,cause the processor to perform the clustering by performing at least oneof hierarchical clustering, K-means clustering, Expectation-Maximizationclustering, or density-based clustering.
 15. The system of claim 11,wherein the instructions, when executed by the processor, cause theprocessor to perform the clustering by using evaluation algorithms toscore and determine a clustering strategy.
 16. The system of claim 11,wherein the instructions, when executed by the processor, cause theprocessor to optimize the importance sampling (IS) distribution functioncorresponding to each of the failure regions by performing at least oneof sequential importance sampling, probability collectives, orregression-based sampling.
 17. The system of claim 11, wherein theinstructions, when executed by the processor, further cause theprocessor to parameterize the mGaussian function.
 18. The system ofclaim 17, wherein the instructions, when executed by the processor,cause the processor to parameterize the mGaussian function by performingat least one of closed-form calculations or convolutional calculations.19. The system of claim 11, wherein the instructions, when executed bythe processor, cause the processor to apply the dimensional reductionmethod by at least one of principal-component-analysis or filtering.